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Gödel's constructible universe L is built by transfinite recursion: L₀ = ∅, L_{α+1} = the set of all subsets of L_α that are definable by a first-order formula with parameters from L_α, and L_λ = ∪_{α<λ} L_α at limits. The axiom V = L asserts that every set is constructible — that the universe of sets equals L. Gödel proved that L is an inner model of ZFC that satisfies both the axiom of choice (AC) and the generalized continuum hypothesis (GCH). This established the relative consistency of AC and GCH with ZF: if ZF is consistent, so is ZFC + GCH. The constructible universe is the 'thinnest' inner model of ZF, containing only those sets that can be explicitly defined at each stage.
Compare the cumulative hierarchy V_α (where V_{α+1} = P(V_α), taking all subsets) with L_α (where L_{α+1} takes only definable subsets). The difference is enormous: V_{ω+1} contains all subsets of ω (uncountably many), while L_{ω+1} contains only countably many (those definable by formulas). Then trace Gödel's argument that L satisfies AC by constructing a definable well-ordering, and that L satisfies GCH because definable power sets are controlled in size.
From the axiom of regularity, you know that every set is built up from ∅ by iterated application of the power set and union operations, stratified into the cumulative hierarchy V_α. Recall: V₀ = ∅, V_{α+1} = P(V_α) (all subsets), V_λ = ∪_{α<λ} V_α at limits, and V = ∪_α V_α. Gödel's constructible universe L is defined by the same transfinite recursion, but with a crucial restriction: instead of taking *all* subsets at each successor stage, you take only the definable ones.
Formally, L₀ = ∅, L_{α+1} = Def(L_α) where Def(X) is the set of all subsets of X definable by a first-order formula with parameters from X, and L_λ = ∪_{α<λ} L_α at limits. The difference is enormous at the first infinite stage: V_{ω+1} = P(ω) contains all subsets of ω, which is an uncountable collection. But L_{ω+1} contains only the subsets of ω that are *first-order definable* from elements of L_ω — and there are only countably many formulas and countably many parameters, so L_{ω+1} is countable. L is the universe you get if you are maximally conservative: only a set that you can explicitly define is allowed to exist.
Gödel proved that L is an inner model of ZFC: it contains all the ordinals, satisfies all the ZFC axioms, and is closed under all set-theoretic operations. The key to proving L satisfies AC is that L carries a canonical definable well-ordering: to well-order all of L, order sets first by the stage L_α they appear in, then by the formula that defines them, then lexicographically by their parameters. This well-ordering is absolute and total, giving AC for free inside L. GCH follows because at each stage L_{α+1} adds only countably many new subsets of L_α (for countable α), keeping the power sets "as small as possible."
The consistency result is profound: if ZF is consistent, then ZF + AC + GCH is consistent, because L is a model. This was Gödel's 1938 result, answering one half of Hilbert's first problem (Cantor's continuum hypothesis). But V = L is not a theorem — it is an additional axiom asserting that every set is constructible. Most set theorists reject V = L as too restrictive: it is incompatible with the existence of measurable cardinals, and more generally with large cardinal axioms, which are widely viewed as the "correct" upper reach of the set-theoretic universe. L is best understood not as the true universe but as a *minimal* model — the most parsimonious universe consistent with ZFC — against which richer universes (those containing large cardinals, or those produced by forcing) are contrasted.